Born theory

B1.6.2.1 BETHE-BORN THEORY FOR HIGH-ENERGY ELECTRON SCATTERING... 

Although the Born equation is a rough approximation, it is often used for comparison of the solvation effects of various solvents. The simplification involved in the Born theory is based primarily on the assumption that the permittivity of the solvent is the same in the immediate vicinity of the ion as in the pure solvent, and the work required to compress the solvent around the ion is neglected. 

Figure 19 The upper portion of the figure shows the effective charge of the He" projectile as a function of the ejected electron energy that is obtained from the experiment [68], Born theory [70], and the model of Toburen et al. [71]. The lower portion of the figure is the same data, but plotted in terms of the impact parameter. The impact parameter is obtained using the relationship between projectile velocity, energy loss, and impact parameter defined by the Massey criterion. (See the text and Ref 71 for details.)...

The upshot is that the Born theory of solvation fails because it regards the solvent as a continuous dielectric, whereas in fact solute ions (especially metal cations with z > 1) often interact in a specific manner with solvent molecules. In any event the molecular dielectric is obviously very lumpy on the scale of the ions themselves. The Born theory and other continuous dielectric models work reasonably well when metal ion solute species are treated as solvent complexes such as Cr(OH2)63+ rather than naked ions such as Cr3+, but the emerging approach to solvation phenomena is to simulate solvation dynamically at the molecular level using computer methods. 

If we consider two solvents of different permittivities (I and II), the pKa value of an acid A in the two solvents should differ by the amount expressed by Eq. (3.16), according to the electrostatic Born theory ... 

Assuming that the electrostatic contributions are given by Born theory and that the solvated ions, irrespective of the composition of the solvation shell, have the same radii, then Equation 19, utilizing the assumptions embodied in Equations 25, 26, and 27, simplifies to... 

Figure 1.11 Solvation energy Usoiv versus cavity radius a solid line corresponds to Equation (1.150) [20] circles to Equation (1.153) [6] dashed line to the Born theory (e0 = 78.39f e

The importance of solvent parameters such as DN and AN and the advantage of their use over physical-electrostatic parameters was further demonstrated by Mayer et al. [21], who studied correlations between the DN and AN of solvents and redox potentials and their chemical equilibrium and ion pair equilibria. According to the Born theory, redox potentials should depend linearly on the reciprocal of the solvent s dielectric constant. Plotting Em values of a redox such as Cd/Cd2+ versus 1/e of the solvents in which it is measured results in a very scattered picture. In contrast, it has been clearly shown by Mayer et al. [15] that redox potentials of metals (e.g., Zn/Zn2+, Cd/Cd2+, Eu/Eu2+) can be nicely... 

The shell model has its origin in the Born theory of lattice dynamics, used in studies of the phonon dispersion curves in crystals/ Although the Born theory includes the effects of polarization at each lattice site, it does not account for the short-range interactions between sites and, most importantly, neglects the effects of this interaction potential on the polarization behavior. The shell model, however, incorporates these short-range interactions. 

Specifically, in the Born theory model of solvation, the intermolecular relaxation energy is (21)... 

The distance between the ions in a crystal is determined by the equilibrium between the forces of attraction and repulsion. Values of the interionic distances may be obtained from x-ray data. On the basis of the Born theory of lattice energies we have,... 

In the present chapter, the properties of electrolyte solutions in water are discussed in detail. Initially the solvation of ions in infinitely dilute solutions is considered on the basis of the Born theory. Then, the Debye-Hiickel model for... 

Limitations of the Born Theory in applications to solvent polarization by ions and its extensions to treatment of kinetics of ionic reactions. 

Found linear relationship between log Kstgb.and 1/D, in accordance with Born theory... 

The shell model has its origin in the Born theory of lattice dynamics, used in studies of the phonon dispersion curves in crystals.70,71 Although the Born theory includes the effects of polarization at each lattice site, it does not account for the short-range interactions between sites and, most importantly, neglects the effects of this interaction potential on the polarization behavior. The shell model, however, incorporates these short-range interactions.72,73 The earliest applications of the shell model, as with the Born model, were to analytical studies of phonon dispersion relations in solids.74 These early applications have been well reviewed elsewhere.71,75-77 In general, lattice dynamics applications of the shell model do not attempt to account for the dynamics of the nuclei and typically use analytical techniques to describe the statistical mechanics of the shells. Although the shell model continues to be used in this fashion,78 lattice dynamics applications are beyond the scope of this chapter. In recent decades, the shell model has come into widespread use as a model Hamiltonian for use in molecular dynamics simulations it is these applications of the shell model that are of interest to us here. 

Szigeti charge does not represent the deviation of ihe nature of the bonds from purely ionic, which is the startii point in the Born theory, but the influence of the approximations made in this theory. 

Figure 2 gives an example of the potential of correlative methods. The curve, calculated with a relatively simple correlative method of [8,9], should be compared to the straightforward extension of conventional. Born theory-based models (area) for the solubiHty of the amino acid L-valine in an alcohol-water mixture (markers). 

Generalized Born Theory for the Study of Structure, Stability, and Interactions of Membrane Proteins. 

First Born theory leads to a differential scattering cross section for loss AE, dc7 = 4( ) S([Pg.550]

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